Homomorphisms and composition operators on algebras of analytic functions of bounded type

Let U and V be convex and balanced open subsets of the Banach spaces X and Y, respectively. In this paper we study the following question: given two Fréchet algebras of holomorphic functions of bounded type on U and V, respectively, that are algebra isomorphic, can we deduce that X and Y (or X* and Y*) are isomorphic? We prove that if X* or Y* has the approximation property and Hwu(U) and Hwu(V) are topologically algebra isomorphic, then X* and Y* are isomorphic (the converse being true when U and V are the whole space). We get analogous results for Hb(U) and Hb(V), giving conditions under which an algebra isomorphism between Hb(X) and Hb(Y) is equivalent to an isomorphism between X* and Y*. We also obtain characterizations of different algebra homomorphisms as composition operators, study the structure of the spectrum of the algebras under consideration and show the existence of homomorphisms on Hb(X) with pathological behaviors.